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Chapter 10 Solids & Liquids continued 10.3 Pascal’s Principle PASCAL’S PRINCIPLE Any change in the pressure applied to a completely enclosed fluid is transmitted undiminished to all parts of the fluid and enclosing walls. F2 F1 P2 = ; P1 = A2 A1 Assume weight of fluid in the tube is negligible P2 = P1 + ρ gh P >> ρ gh P2 = P1 Small ratio F2 F1 = ⇒ A2 A1 ⎛ A1 ⎞ F1 = F2 ⎜ ⎟ ⎝ A2 ⎠ 10.4 Archimedes’ Principle Liquid density m ρ (rho) = V P2 = P1 + ρ gh Pressure grows linearly with depth. F1 = F2 = Buoyant Force, FB P1 FB = F2 + (− F1 ) P2 = ρ ghA = P2 A − P1 A = ( P2 − P1 ) A = ρV since P2 − P1 = ρ gh g and V = hA mass of displaced fluid Buoyant force = Weight of displaced fluid FB = ρVg 10.4 Archimedes’ Principle ARCHIMEDES’ PRINCIPLE Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces: CORROLARY If an object is floating then the magnitude of the buoyant force is equal to the magnitude of its weight. 10.4 Archimedes’ Principle Any fluid applies a buoyant force to an object that is partially or completely immersed in it; the magnitude of the buoyant force equals the weight of the fluid that the object displaces: FB = ρ WaterVg W = ρObjectVg FB = W − T (Net force = 0) FB T = 1− W W ρ WaterVg T = 1− ρObjectVg W ρObject ρ Water = (1− T / W ) 10.4 Archimedes’ Principle Use Archimedes Principle to measure the density of Aluminum ρObject ρ Water = (1− T / W ) T 4.5N ~ = 0.53 W 8.5N 1000 kg/m 3 = = 2100 kg/m 3 1− 0.53 10.4 Archimedes’ Principle Example: A Swimming Raft The raft is made of solid square pinewood. a) Determine whether the raft floats in water and if so, b) how much of the raft is beneath the surface. 10.4 Archimedes’ Principle Raft has these properties If Wraft < FB submerged, raft floats Wraft Weight of the Raft = mraft g = ρraftVraft g ( )( Vraft = ( 4.0 ) ( 4.0 ) ( 0.30 ) m 3 = 4.8 m 3 ρraft = 550 kg/m 3 )( = 550kg m 3 4.8m 3 9.80m s 2 ) = 26000 N Buoyant force totally submerged FB = ρ Vg = ρWaterVRaft g ( )( )( = 1000kg m 3 4.8m 3 9.80m s 2 ) = 47000 N Wraft < FB submerged, so Raft floats Only part of the raft is above water 10.4 Archimedes’ Principle Raft has these properties Vraft = ( 4.0 ) ( 4.0 ) ( 0.30 ) m 3 = 4.8 m 3 ρraft = 550 kg/m 3 An object will float in a fluid if its density is less than that of the fluid ! ρ Raft = 550kg/m 3 < 1000kg/m 3 = ρWater So, how much of the raft is beneath the surface? Only part of the raft is above water 10.4 Archimedes’ Principle How much of raft below water? For a floating object FB = WRaft = 26000 N FB is weight of displaced water FB = ρ Water gVWater (displaced) = ρ Water g( ARaft h) WRaft h= (height below surface) ρ Water gARaft = 26000N (1000kg m )(9.80m s )(16.0 m ) = 0.17 m 3 2 2 10.4 Archimedes’ Principle In General What fraction of full height H is below the surface ? WRaft h= ρ Water gARaft ρ Raft gARaft H = ρ Water gARaft ρ Raft h = fraction below surface H ρ Water Fraction below surface is just the ratio of the density of floating object to the density of the fluid. H = 0.30 m 10.5 Fluids in Motion In steady flow the velocity of the fluid particles at any point is constant as time passes. Unsteady flow exists whenever the velocity of the fluid particles at a point changes as time passes. Turbulent flow is an extreme kind of unsteady flow in which the velocity of the fluid particles at a point change erratically in both magnitude and direction. Fluid flow can be compressible or incompressible. Most liquids are nearly incompressible. Fluid flow can be viscous or nonviscous. An incompressible, nonviscous fluid is called an ideal fluid. 10.5 Fluids in Motion When the flow is steady, streamlines are often used to represent the trajectories of the fluid particles. The mass of fluid per second that flows through a tube is called the mass flow rate. 10.5 The Equation of Continuity EQUATION OF CONTINUITY The mass flow rate has the same value at every position along a tube that has a single entry and a single exit for fluid flow. SI Unit of Mass Flow Rate: kg/s Δm2 = ρ2 A2 v2 Δt Incompressible fluid: Volume flow rate Q: ρ1 = ρ2 10.5 The Equation of Continuity Example: A Garden Hose A garden hose has an unobstructed opening with a cross sectional area of 2.85x10-4m2. It fills a bucket with a volume of 8.00x10-3m3 in 30 seconds. Find the speed of the water that leaves the hose through (a) the unobstructed opening and (b) an obstructed opening with half as much area. a) Q = Av Q v= = A b) ( ) 8.00 × 10−3 m 3 / ( 30.0 s ) 2.85 × 10 m -4 2 = 0.936m s A1v1 = A2 v2 A1 v2 = v1 = ( 2 ) ( 0.936m s ) = 1.87 m s A2 10.5 Bernoulli’s Equation The fluid accelerates toward the lower pressure regions. According to the pressure-depth relationship, the pressure is lower at higher levels, provided the area of the pipe does not change. Apply Work-Energy theorem to determine relationship between pressure, height, velocity, of the fluid. 10.5 Bernoulli’s Equation Work done by tiny pressure “piston” WΔP = (∑ F ) s = ( ΔF ) s = ( ΔPA) s; V = As Work (NC) done by pressure difference from 2 to 1 WNC = ( P2 − P1 )V E1 = 12 mv12 + mgy1 E2 = 12 mv22 + mgy2 WNC = E1 − E2 = ( 1 2 ) ( mv12 + mgy1 − 1 2 mv22 + mgy2 ) 10.5 Bernoulli’s Equation WNC = ( P2 − P1 )V WNC = E1 − E2 = ( 1 2 ) ( mv + mgy1 − 2 1 1 2 mv + mgy2 2 2 ) NC Work yields a total Energy change. Equating the two expressions for the work done, ( P − P )V = ( 2 1 (P − P) = ( 2 1 1 2 1 2 ) ( mv12 + mgy1 − ) ( ρ v12 + ρ gy1 − 1 2 1 2 mv22 + mgy2 ρ v22 + ρ gy2 ) m = ρV ) Rearrange to obtain Bernoulli's Equation BERNOULLI’S EQUATION In steady flow of a nonviscous, incompressible fluid, the pressure, the fluid speed, and the elevation at two points are related by: P1 + 12 ρ v12 + ρ gy1 = P2 + 12 ρ v22 + ρ gy2 10.5 Applications of Bernoulli’s Equation Conceptual Example: Tarpaulins and Bernoulli’s Equation When the truck is stationary, the tarpaulin lies flat, but it bulges outward when the truck is speeding down the highway. Account for this behavior. Relative to moving truck v1 = 0 under the tarp Bernoulli’s Equation P1 + ρ v + ρ gy1 = P2 + ρ v + ρ gy2 1 2 2 1 1 2 P1 = P2 + 12 ρ v22 P1 > P2 2 2 v2 air flow over top 10.5 Applications of Bernoulli’s Equation 10.5 Applications of Bernoulli’s Equation Example: Efflux Speed The tank is open to the atmosphere at the top. Find and expression for the speed of the liquid leaving the pipe at the bottom. P1 = P2 = Patmosphere (1× 105 N/m 2 ) v2 = 0, y2 = h, y1 = 0 P1 + 12 ρ v12 + ρ gy1 = P2 + 12 ρ v22 + ρ gy2 1 2 ρ v12 = ρ gh v1 = 2gh 10.6 Viscous Flow Flow of an ideal fluid. Flow of a viscous fluid. FORCE NEEDED TO MOVE A LAYER OF VISCOUS FLUID WITH CONSTANT VELOCITY The magnitude of the tangential force required to move a fluid layer at a constant speed is given by: η Av F= y η , is the coefficient of viscosity SI Unit: Pa ⋅s; 1 poise (P) = 0.1 Pa ⋅s POISEUILLE’S LAW (flow of viscous fluid) The volume flow rate is given by: Pressure drop in a straight uniform diamater pipe. 11.11 Viscous Flow Example: Giving and Injection A syringe is filled with a solution whose viscosity is 1.5x10-3 Pa·s. The internal radius of the needle is 4.0x10-4m. The gauge pressure in the vein is 1900 Pa. What force must be applied to the plunger, so that 1.0x10-6m3 of fluid can be injected in 3.0 s? 8η LQ P2 − P1 = π R4 8 1.5 × 10−3 Pa ⋅s ( 0.025 m ) 1.0 × 10−6 m 3 3.0 s = = 1200 Pa 4 π 4.0 × 10-4 m ( ) ( ( ) P2 = (1200 + P1 ) Pa = (1200 + 1900 ) Pa = 3100 Pa ( ) F = P2 A = ( 3100 Pa ) 8.0 × 10−5 m 2 = 0.25N )